Shape Optimization of Coils

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Shape Optimization of Coils

This model demonstrates how to design a coil geometry using gradient-based shape optimization. The objective is to achieve a uniform magnetic field along the coil axis and a field minimum near the axis ends. The model assumes azimuthal symmetry, but one can use the optimization result as inspiration for a 3D design.

Model Definition

The model is set up with 6 coils near the center and one coil at each end with the opposite current direction. The current in the coils at the ends is also optimized to mimic the effect of having coils with partial turns. The result of the shape optimization is shown in Figure 2, while the initial geometry for this model is shown in Figure 1.

Figure 1: The initial geometry.

The and features are used to allow the coils to move in the radial direction. There are 8 control variables for the coil positions and one for the current in the outer coils. The IPOPT optimization can solve such problems fast.

The objective function,

, consists of two terms:

Results and Discussion

Figure 2: The optimized geometry.

Figure 2 shows the optimized design. The first and second objectives are reduced by around 90% and 50%, respectively. The strength of the magnetic field on the axis is illustrated in Figure 3.

Figure 3: The optimization increases the magnetic field in the middle domain and decreases it in the outer domains.

Notes About the COMSOL Implementation

This model can be constructed in a way that exploits symmetry in the xy-plane, leading to a reduction in the computational time.

Optimization_Module/Shape_Optimization/coil_shape_optimization

Modeling Instructions

From the menu, choose .

In the window, click

.

1

In the window, click

.

2

In the tree, select .

3

4

5

In the tree, select .

6

1

In the window, expand the node.

2

Right-click and choose .

3

Browse to the model’s Application Libraries folder and double-click the file coil_shape_optimization_geom_sequence.mph.

4

In the toolbar, click

.

5

button in the toolbar.

The geometry should now look like that in Figure 1.

Global Definitions

Geometry Parameters

1

In the window, under click .

2

In the window for , type Geometry Parameters in the text field.

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Name	Expression	Value	Description
f0	1[kHz]	1000 Hz	Frequency
lastTurns	0.5	0.5	Outer loop current factor
dmax	3[cm]	0.03 m	Maximum coil translation

1

In the toolbar, click

to open the window.

2

Go to the window.

3

In the tree, select .

4

Click in the window toolbar.

5

In the tree, select .

6

Click in the window toolbar.

7

In the toolbar, click

to close the window.

1

In the window for , locate the section.

2

From the list, choose .

Magnetic Fields (mf)

1

In the window, under right-click and choose the domain setting .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. Select the check box.

5

In the Icoil text field, type 1[kA].

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the Icoil text field, type 1[kA]*lastTurns.

Setting a coil with a fraction of the current is a way to take into account of a fraction of a full turn in 2D a axisymmetric model; similarly a current with opposite sign may represent the same a coil that is wounded in the opposite direction.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the Icoil text field, type -1[kA]*lastTurns.

1

In the toolbar, click

.

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

Free Triangular 1

In the toolbar, click

.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. From the list, choose .

6

.

In the toolbar, click

and choose .

Shape Optimization

Free Shape Domain 1

1

In the window for , locate the section.

2

From the list, choose .

Transformation 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the table, enter the following settings:


	Lock	Lower bound (m)	Upper bound (m)
R		-dmax	dmax
Z		-coilSpace/4	coilSpace/4

5

Locate the section. From the list, choose .

Average inner magnetic field

1

In the toolbar, click

and choose .

2

In the window for , type Average inner magnetic field in the text field.

3

In the text field, type Bavg.

4

Locate the section. From the list, choose .

Average squared deviation from average

1

Right-click and choose .

2

In the window for , type Average squared deviation from average in the text field.

3

In the text field, type Bdev_sq.

4

Locate the section. In the text field, type (mf.normB-Bavg)^2.

Average outer magnetic field

1

Right-click and choose .

2

In the window for , type Average outer magnetic field in the text field.

3

In the text field, type Bouter.

4

Locate the section. From the list, choose .

5

Locate the section. In the text field, type mf.normB.

Infinite Element Domain 1 (ie1)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

From the list, choose .

Step 1: Frequency Domain

1

In the window, under click .

2

In the window for , locate the section.

3

In the text field, type f0.

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Find the subsection. From the list, choose .

5

In the text field, type 1000.

This effectively saves the first and last iteration.

6

Locate the section. From the list, choose .

7

Click in the upper-right corner of the section. From the menu, choose 4.

8

Click in the upper-right corner of the section. From the menu, choose .

Take the square root of the Bdev_sq, so that the two objectives get the same unit.

9

Locate the section. In the table, enter the following settings:


Expression	Description	Evaluate for
sqrt(comp1.Bdev_sq)	Domain Probe 2	Frequency Domain

10

Locate the section. Click

.

Initialize the study to generate a plot for use while solving.

11

In the table, enter the following settings:


Parameter name	Initial value	Scale	Lower bound	Upper bound
lastTurns (Outer loop current factor)	0.5	1	0	1

12

Locate the section. From the list, choose .

13

In the window, click .

14

In the window for , type Shape Optimization in the text field.

15

In the toolbar, click

.

Shape Optimization

The plot shows the displacement. Add an feature to also show the value of the lastTurns parameter.

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

In the text field, type eval(lastTurns).

4

Locate the section. Clear the check box.

5

From the list, choose .

Shape Optimization

1

In the window, under click .

2

In the window for , locate the section.

3

Select the check box.

4

From the list, choose .

Enable move limits to reduce the risk of inverted elements.

Solver Configurations

In the window, expand the node.

Solution 1 (sol1)

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

Select the check box.

4

Select the check box.

5

In the associated text field, type 50.

6

Magnetic Flux Density Norm, Revolved Geometry (mf)

1

In the toolbar, click

.

2

button in the toolbar.

1

In the toolbar, click

and choose .

2

In the window for , type Field on Axis in the text field.

3

Locate the section. From the list, choose .

4

Click to expand the section. From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

5

In the text field, type z.

6

Click to expand the section. Select the check box.

7

From the list, choose .

8

In the table, enter the following settings:


Legends
Initial Design

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. In the table, enter the following settings:


Legends
Optimized Design

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

5

In the text field, type z.

6

Click to expand the section. From the list, choose .

7

In the text field, type 3.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. From the list, choose .

Line Graph 3, Line Graph 4

1

In the window, under , Ctrl-click to select and .

2

Right-click and choose .

1

In the window for , locate the section.

2

From the list, choose .

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

4

In the toolbar, click

.

5

button in the toolbar.

Compute the value of the objective functions before and after optimization.

Evaluation Group 1

In the toolbar, click

.

Global Evaluation 1

1

Right-click and choose .

2

In the window for , click in the upper-right corner of the section. From the menu, choose 4.

3

Click in the upper-right corner of the section. From the menu, choose .

4

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
sqrt(Bdev_sq)	T
Bouter	T	Average outer magnetic field

5

In the toolbar, click

.

Compute the value of the corners, so that one can construct an interpolation function for 3D verification.

Point Displacements

1

In the toolbar, click

.

2

In the window for , type Point Displacements in the text field.

3

Locate the section. From the list, choose .

Point Evaluation 1

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
r	m	r-coordinate
z	m	z-coordinate

5

In the toolbar, click

.

Geometry Modeling Instructions

From the menu, choose .

In the window, click

.

1

In the window, click

.

2

In the tree, select .

3

4

5

In the tree, select .

6

1

In the window, expand the node.

2

Right-click and choose .

3

Browse to the model’s Application Libraries folder and double-click the file coil_shape_optimization_geom_sequence.mph.

4

In the toolbar, click

.

5

button in the toolbar.

The geometry should now look like that in Figure 1.

Global Definitions

Geometry Parameters

1

In the window, under click .

2

In the window for , type Geometry Parameters in the text field.

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Name	Expression	Value	Description
f0	1[kHz]	1000 Hz	Frequency
lastTurns	0.5	0.5	Outer loop current factor
dmax	3[cm]	0.03 m	Maximum coil translation

1

In the toolbar, click

to open the window.

2

Go to the window.

3

In the tree, select .

4

Click in the window toolbar.

5

In the tree, select .

6

Click in the window toolbar.

7

In the toolbar, click

to close the window.

1

In the window for , locate the section.

2

From the list, choose .

Magnetic Fields (mf)

1

In the window, under right-click and choose the domain setting .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. Select the check box.

5

In the Icoil text field, type 1[kA].

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the Icoil text field, type 1[kA]*lastTurns.

Setting a coil with a fraction of the current is a way to take into account of a fraction of a full turn in 2D a axisymmetric model; similarly a current with opposite sign may represent the same a coil that is wounded in the opposite direction.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the Icoil text field, type -1[kA]*lastTurns.

1

In the toolbar, click

.

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

Free Triangular 1

In the toolbar, click

.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. From the list, choose .

6

.

In the toolbar, click

and choose .

Shape Optimization

Free Shape Domain 1

1

In the window for , locate the section.

2

From the list, choose .

Transformation 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the table, enter the following settings:


	Lock	Lower bound (m)	Upper bound (m)
R		-dmax	dmax
Z		-coilSpace/4	coilSpace/4

5

Locate the section. From the list, choose .

Average inner magnetic field

1

In the toolbar, click

and choose .

2

In the window for , type Average inner magnetic field in the text field.

3

In the text field, type Bavg.

4

Locate the section. From the list, choose .

Average squared deviation from average

1

Right-click and choose .

2

In the window for , type Average squared deviation from average in the text field.

3

In the text field, type Bdev_sq.

4

Locate the section. In the text field, type (mf.normB-Bavg)^2.

Average outer magnetic field

1

Right-click and choose .

2

In the window for , type Average outer magnetic field in the text field.

3

In the text field, type Bouter.

4

Locate the section. From the list, choose .

5

Locate the section. In the text field, type mf.normB.

Infinite Element Domain 1 (ie1)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

From the list, choose .

Step 1: Frequency Domain

1

In the window, under click .

2

In the window for , locate the section.

3

In the text field, type f0.

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Find the subsection. From the list, choose .

5

In the text field, type 1000.

This effectively saves the first and last iteration.

6

Locate the section. From the list, choose .

7

Click in the upper-right corner of the section. From the menu, choose 4.

8

Click in the upper-right corner of the section. From the menu, choose .

Take the square root of the Bdev_sq, so that the two objectives get the same unit.

9

Locate the section. In the table, enter the following settings:


Expression	Description	Evaluate for
sqrt(comp1.Bdev_sq)	Domain Probe 2	Frequency Domain

10

Locate the section. Click

.

Initialize the study to generate a plot for use while solving.

11

In the table, enter the following settings:


Parameter name	Initial value	Scale	Lower bound	Upper bound
lastTurns (Outer loop current factor)	0.5	1	0	1

12

Locate the section. From the list, choose .

13

In the window, click .

14

In the window for , type Shape Optimization in the text field.

15

In the toolbar, click

.

Shape Optimization

The plot shows the displacement. Add an feature to also show the value of the lastTurns parameter.

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

In the text field, type eval(lastTurns).

4

Locate the section. Clear the check box.

5

From the list, choose .

Shape Optimization

1

In the window, under click .

2

In the window for , locate the section.

3

Select the check box.

4

From the list, choose .

Enable move limits to reduce the risk of inverted elements.

Solver Configurations

In the window, expand the node.

Solution 1 (sol1)

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

Select the check box.

4

Select the check box.

5

In the associated text field, type 50.

6

Magnetic Flux Density Norm, Revolved Geometry (mf)

1

In the toolbar, click

.

2

button in the toolbar.

1

In the toolbar, click

and choose .

2

In the window for , type Field on Axis in the text field.

3

Locate the section. From the list, choose .

4

Click to expand the section. From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

5

In the text field, type z.

6

Click to expand the section. Select the check box.

7

From the list, choose .

8

In the table, enter the following settings:


Legends
Initial Design

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. In the table, enter the following settings:


Legends
Optimized Design

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

5

In the text field, type z.

6

Click to expand the section. From the list, choose .

7

In the text field, type 3.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. From the list, choose .

Line Graph 3, Line Graph 4

1

In the window, under , Ctrl-click to select and .

2

Right-click and choose .

1

In the window for , locate the section.

2

From the list, choose .

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

4

In the toolbar, click

.

5

button in the toolbar.

Compute the value of the objective functions before and after optimization.

Evaluation Group 1

In the toolbar, click

.

Global Evaluation 1

1

Right-click and choose .

2

In the window for , click in the upper-right corner of the section. From the menu, choose 4.

3

Click in the upper-right corner of the section. From the menu, choose .

4

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
sqrt(Bdev_sq)	T
Bouter	T	Average outer magnetic field

5

In the toolbar, click

.

Compute the value of the corners, so that one can construct an interpolation function for 3D verification.

Point Displacements

1

In the toolbar, click

.

2

In the window for , type Point Displacements in the text field.

3

Locate the section. From the list, choose .

Point Evaluation 1

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
r	m	r-coordinate
z	m	z-coordinate

5

In the toolbar, click

.